# Understanding Eigenvalues and Eigenvectors: Exploring a Matrix Transformation

• rmiller70015
In summary, the conversation discusses finding the eigenvalues and eigenvectors of a given matrix. The student initially makes a mistake in their calculations and presents incorrect eigenvectors, which are then corrected by another person in the conversation. The correct eigenvalues are 0 and 8, and the corresponding eigenvectors are [-2, 1] and [2, 3].
rmiller70015

## Homework Statement

$$\frac{d\vec{Y}}{dt} = \begin{bmatrix} 2 & 4 \\ 3 & 6 \end{bmatrix} \vec{Y}$$
Find the eigenvalues and eigenvectors

## The Attempt at a Solution

I found the eigenvectors to be
$$\vec{v_1} = \begin{bmatrix} 2 \\ 1 \end{bmatrix} , \vec{v_2} = \begin{bmatrix} 2 \\ -3 \end{bmatrix}$$

I found a widget on Wolfram Alpha that says the second eigenvector should be:
$$\begin{bmatrix} 2 \\ 3 \end{bmatrix}$$

I am more inclined to believe wolfram alpha is correct, but can someone show me why?

How are we to determine whether, and if so how, you have made an error if you don't post your working?

It was mostly by looking at the matrix and trying to make them the same and now that you say that when I do the algebra on it to post it, I see what I did wrong. I just tried to look at 3 and -2 to find the solution when I should have said 3x-2y=0 and 3x=2y then the vector becomes clear.

rmiller70015 said:
It was mostly by looking at the matrix and trying to make them the same and now that you say that when I do the algebra on it to post it, I see what I did wrong. I just tried to look at 3 and -2 to find the solution when I should have said 3x-2y=0 and 3x=2y then the vector becomes clear.

Neither of your ##v_1## or ##v_2## are eigenvectors. If ##A## is your matrix we have ##Av_1 = (8,12)^T##, which cannot be a multiple of ##v_1##: the first component of ##v_1## is larger than the second component, while the opposite is true for ##Av_1##. Similarly, ##v_2## cannot be an eigenvector of ##A## because the components of ##v_2## have opposite signs, while those of ##Av_2## have the same sign.

Also: you did not show the eigenvalues.

rmiller70015
Eigens are 0,8. The problem was v2 I ommited a negative sign in the first vector on accident.

rmiller70015 said:
Eigens are 0,8. The problem was v2 I ommited a negative sign in the first vector on accident.

I do not understand what you are trying to say. What is the correct ##v_1##? What is the correct ##v_2##? Instead of trying to describe these in words, just show the actual numerical entries.

rmiller70015
$$\vec{v_1}= \begin{bmatrix} -2 \\ 1 \end{bmatrix} , \vec{v_2}= \begin{bmatrix} 2 \\ 3 \end{bmatrix}$$

Sory for putting so little work into this, I've got the answer and moved onto the next problem at this point.

## What are real and zero eigenvalues?

Real and zero eigenvalues are values that are associated with a square matrix in linear algebra. They represent the values that, when multiplied by the matrix, result in the same vector but with a different scalar value. Real eigenvalues have a non-zero scalar value, while zero eigenvalues have a scalar value of 0.

## How do I find the real and zero eigenvalues of a matrix?

To find the real and zero eigenvalues of a matrix, you can use the eigenvalue equation: Ax = λx. Here, A is the matrix, x is the vector, and λ is the eigenvalue. You can solve this equation by finding the determinant of the matrix and setting it equal to 0, then solving for λ.

## What is the significance of real and zero eigenvalues?

The real and zero eigenvalues of a matrix have significant implications in linear algebra and other fields of science and mathematics. They are used to find the eigenvectors of a matrix, which can be used to simplify complicated systems of equations and understand the behavior of linear transformations.

## Can a matrix have both real and zero eigenvalues?

Yes, a matrix can have both real and zero eigenvalues. In fact, most matrices have a combination of both real and zero eigenvalues. This is because the eigenvalues of a matrix depend on its size, shape, and the values of its elements.

## How do I know if a matrix has only real or only zero eigenvalues?

If a matrix has only real eigenvalues, this means that all of its eigenvalues have a non-zero scalar value. If a matrix has only zero eigenvalues, this means that all of its eigenvalues have a scalar value of 0. You can determine this by calculating the eigenvalues of the matrix using the eigenvalue equation.

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